Operating with Rational Expressions - Algebra 2

1. What restriction must be stated for the rational expression $\frac{P(x)}{Q(x)}$?

   $Q(x)\neq 0$ (exclude values making $Q(x)=0$). The denominator cannot equal zero to avoid division by zero.

2. What value must be excluded when simplifying $\frac{(x-3)(x+1)}{x-3}$?

   $x\neq 3$. The factor $(x-3)$ in original expression cannot equal zero.

3. What is $\frac{x}{x-1}\div\frac{x+2}{x-1}$ simplified?

   $\frac{x}{x+2}$. Multiply by reciprocal and cancel $(x-1)$.

4. What is the rule for dividing by a rational expression $\frac{P(x)}{Q(x)}$?

   Multiply by the reciprocal $\frac{Q(x)}{P(x)}$, with $P(x)\neq 0$. Division by a fraction means multiplying by its reciprocal.

5. What is the result of adding $\frac{a}{b}+\frac{c}{b}$ when denominators match?

   $\frac{a+c}{b}$. Combine numerators over the common denominator.

6. What is $\frac{1}{x}+\frac{1}{x}$ simplified?

   $\frac{2}{x}$. Add fractions with the same denominator by adding numerators.

7. What is $\frac{(x-3)(x+1)}{x-3}$ simplified (as an expression)?

   $x+1$. Cancel the common factor $(x-3)$ from both parts.

8. What is the domain restriction for $\frac{x^2}{(x-1)(x+4)}$?

   $x\neq 1$ and $x\neq -4$. Each factor in the denominator cannot equal zero.

9. What is $\frac{3}{x}-\frac{1}{x}$ simplified?

   $\frac{2}{x}$. Subtract fractions with same denominator: $ (3-1)/x $.

10. What is $\frac{x^2-9}{x-3}\cdot\frac{1}{x+3}$ simplified?

    $1$. Factor $x^2-9=(x+3)(x-3)$ and cancel both factors.

11. What is $\frac{1}{x+2}-\frac{1}{x}$ simplified?

    $\frac{-2}{x(x+2)}$. Find LCD $x(x+2)$ and subtract: $\frac{x-(x+2)}{x(x+2)}$.

12. What is $\frac{x^2-1}{x+1}\cdot\frac{1}{x-1}$ simplified?

    $1$. Factor $x^2-1=(x+1)(x-1)$ and cancel both factors.

13. What is the least common denominator (LCD) of $\frac{1}{x}$ and $\frac{1}{x-2}$?

    $x(x-2)$. Use the product of distinct linear factors.

14. What is $\frac{1}{x}+\frac{1}{x-2}$ simplified?

    $\frac{2x-2}{x(x-2)}$. Find common denominator $x(x-2)$ and add numerators.

15. What is $\frac{1}{x^2-1}-\frac{1}{x+1}$ simplified?

    $\frac{-x}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{1-(x-1)}{(x-1)(x+1)}$.

16. What is $\frac{x}{x+2}+\frac{2}{x+2}$ simplified?

    $\frac{x+2}{x+2}=1$. Add numerators over common denominator: $(x+2)/(x+2)$.

17. What is $\frac{(x-2)(x+5)}{(x-2)(x-1)}$ simplified (keep restrictions implied)?

    $\frac{x+5}{x-1}$. Cancel $(x-2)$ from numerator and denominator.

18. What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?

    $\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.

19. What is $\frac{x}{x-1}-\frac{1}{x-1}$ simplified?

    $\frac{x-1}{x-1}=1$. Subtract numerators over common denominator: $(x-1)/(x-1)$.

20. What is the domain restriction for $\frac{1}{x-5}$?

    $x\neq 5$. Set $x-5=0$ to find where the denominator is undefined.

21. What operation is valid to simplify $\frac{(x-2)(x+5)}{(x-2)(x-1)}$?

    Cancel the common factor $(x-2)$, with $x\neq 2$. Common factors can be cancelled when not equal to zero.

22. What is $\frac{x}{x^2}-\frac{1}{x}$ simplified (for $x\neq 0$)?

    $0$. Rewrite as $\frac{1}{x}-\frac{1}{x}=0$.

23. What is $\frac{x}{x+1}+\frac{1}{x+1}$ simplified?

    $\frac{x+1}{x+1}=1$. Add numerators over common denominator: $(x+1)/(x+1)$.

24. What is $\frac{1}{x}\div\frac{1}{x+5}$ simplified?

    $\frac{x+5}{x}$. Multiply by reciprocal: $\frac{1}{x} \cdot \frac{x+5}{1}$.

25. What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?

    $(x-1)(x+1)$. Since $x^2-1=(x-1)(x+1)$, LCD includes both factors.

26. What is $\frac{2}{x}-\frac{3}{x}$ simplified?

    $\frac{-1}{x}$. Same denominator subtraction: $(2-3)/x = -1/x$.

27. What is $\frac{2}{x+1}+\frac{3}{x-1}$ simplified?

    $\frac{5x+1}{(x+1)(x-1)}$. Use LCD and add: $\frac{2(x-1)+3(x+1)}{(x+1)(x-1)}$.

28. What is the first step to simplify a rational expression?

    Factor numerator and denominator completely. Factoring reveals common factors that can be cancelled.

29. What is a rational expression in terms of polynomials $P(x)$ and $Q(x)$?

    $\frac{P(x)}{Q(x)}$ with $Q(x)\neq 0$. A rational expression is the quotient of two polynomials.

30. What is $\frac{5}{x-4}+\frac{1}{x-4}$ simplified?

    $\frac{6}{x-4}$. Add numerators with same denominator: $ (5+1)/(x-4) $.

31. What is $\frac{1}{x-1}+\frac{1}{x+1}$ simplified?

    $\frac{2x}{(x-1)(x+1)}$. Use LCD $(x-1)(x+1)$ and add: $(x+1+x-1)$.

32. What additional restriction occurs when dividing by $\frac{x-2}{x+1}$?

    $\frac{x-2}{x+1}\neq 0$, so $x\neq 2$ (and also $x\neq -1$). The divisor cannot equal zero for division to be defined.

33. What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?

    $\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.

34. What is $\frac{x}{x^2-1}+\frac{1}{x+1}$ simplified?

    $\frac{2x+1}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{x+(x-1)}{(x-1)(x+1)}$.

35. What is $\frac{x-3}{x}\div\frac{x-3}{x+1}$ simplified (as an expression)?

    $\frac{x+1}{x}$. Cancel common factor $(x-3)$ after multiplying by reciprocal.

36. What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?

    $1$. Rewrite division as multiplication: $\frac{x^2-1}{x+1} \cdot \frac{1}{x-1}$.

37. What does it mean for rational expressions to be closed under addition and subtraction?

    The sum or difference of rational expressions is rational. The result of adding or subtracting is always a rational expression.

38. What is the domain restriction for $\frac{2}{x^2+1}$ over the real numbers?

    No real restrictions (since $x^2+1\neq 0$ for real $x$). $x^2+1>0$ for all real $x$, so no denominator zeros exist.

39. What is $\frac{1}{x}-\frac{1}{x+1}$ simplified?

    $\frac{1}{x(x+1)}$. Find LCD $x(x+1)$ and subtract: $\frac{x+1-x}{x(x+1)}$.

40. What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?

    $(x-1)^2$. Use the highest power of each factor.

41. What is the result of subtracting $\frac{a}{b}-\frac{c}{b}$ when denominators match?

    $\frac{a-c}{b}$. Subtract numerators over the common denominator.

42. What is $\frac{x}{x-1}\cdot\frac{x-1}{x+2}$ simplified?

    $\frac{x}{x+2}$. Cancel the common factor $(x-1)$.

43. What is $\frac{1}{x+1}-\frac{1}{x-1}$ simplified?

    $\frac{-2}{(x+1)(x-1)}$. Use LCD $(x+1)(x-1)$ and subtract: $(x-1-x-1)$.

44. What is $\frac{2}{x}\div\frac{3}{x+1}$ simplified?

    $\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x} \cdot \frac{x+1}{3}$.

45. What is the LCD of $\frac{1}{x^2}$ and $\frac{1}{x^3}$?

    $x^3$. Use the highest power of $x$ among the denominators.

46. What is the LCD of $\frac{1}{x(x+1)}$ and $\frac{1}{x+1}$?

    $x(x+1)$. The LCD is the product of all distinct factors.

47. What is $\frac{2}{x+3}+\frac{1}{x+3}$ simplified?

    $\frac{3}{x+3}$. Add numerators with same denominator: $(2+1)/(x+3)$.

48. What is $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?

    $\frac{6}{x(x+1)}$. Multiply straight across: $(2\cdot 3)/(x(x+1))$.

49. What is $\frac{x}{x^2}+\frac{1}{x}$ simplified (for $x\neq 0$)?

    $\frac{2}{x}$. Rewrite as $\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$.

50. What is the general multiplication rule for $\frac{a}{b}\cdot\frac{c}{d}$?

    $\frac{ac}{bd}$ (with $b\neq 0$ and $d\neq 0$). Multiply numerators and denominators separately.

51. What does it mean for rational expressions to be closed under division?

    The quotient is rational if dividing by a nonzero rational expression. Division by zero is excluded, but the result remains rational.

52. What is $\frac{x}{x^2}\cdot\frac{x^3}{x}$ simplified (for $x\neq 0$)?

    $x$. Simplify powers: $\frac{x \cdot x^3}{x^2 \cdot x} = \frac{x^4}{x^3} = x$.

53. What is the domain restriction for $\frac{x+2}{x^2-9}$?

    $x\neq 3$ and $x\neq -3$. Factor $x^2-9=(x-3)(x+3)$ and exclude where factors equal zero.

54. What does it mean for rational expressions to be closed under multiplication?

    The product of rational expressions is rational. The result of multiplying rational expressions is rational.

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